Question: Simplify and expand the following expression: $ \dfrac{5z - 2}{z - 4}+\dfrac{-8}{z - 6} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(z - 4)(z - 6)$ Multiply the first term by $\dfrac{z - 6}{z - 6}$ $ \begin{align*} \dfrac{5z - 2}{z - 4} \times \dfrac{z - 6}{z - 6} & = \dfrac{(5z - 2)(z - 6)}{(z - 4)(z - 6)} \\ & = \dfrac{5z^2 - 32z + 12}{(z - 4)(z - 6)}\end{align*} $ Multiply the second term by $\dfrac{z - 4}{z - 4}$ $ \begin{align*} \dfrac{-8}{z - 6} \times \dfrac{z - 4}{z - 4} & = \dfrac{(-8)(z - 4)}{(z - 6)(z - 4)} \\ & = \dfrac{-8z + 32}{(z - 6)(z - 4)}\end{align*} $ Now we have: $ = \dfrac{5z^2 - 32z + 12}{(z - 4)(z - 6)} + \dfrac{-8z + 32}{(z - 6)(z - 4)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{5z^2 - 32z + 12 - 8z + 32}{(z - 4)(z - 6)} $ $ = \dfrac{5z^2 - 40z + 44}{(z - 4)(z - 6)}$ Expand the denominator: $ = \dfrac{5z^2 - 40z + 44}{z^2 - 10z + 24}$